Dispersion in time and space: what propagating optical pulses in time (& not space) forces us to confront

TL;DR

This paper introduces a temporally propagated optical pulse equation that preserves causality and offers advantages over traditional spatial models, especially in the few-cycle regime, highlighting differences in dispersion treatment.

Contribution

It derives a new temporally propagated uni-directional optical pulse equation valid in the few cycle limit, enabling direct comparison with bi-directional models and revealing dispersion approximation issues.

Findings

Temporal propagation preserves causality better than spatial models.

The derived equations can be approximated to a uni-directional form for small changes per cycle.

Differences in dispersion handling highlight limitations of spatial propagation approaches.

Abstract

I derive a temporally propagated uni-directional optical pulse equation valid in the few cycle limit. Temporal propagation is advantageous because it naturally preserves causality, unlike the competing spatially propagated models. The exact coupled bi-directional equations that this approach generates can be efficiently approximated down to a uni-directional form in cases where an optical pulse changes little over one optical cycle. They also permit a direct term-to-term comparison of the exact bi-directional theory with its corresponding approximate uni-directional theory. Notably, temporal propagation handles dispersion in a different way, and this difference serves to highlight existing approximations inherent in spatially propagated treatments of dispersion. Accordingly, I emphasise the need for future work in clarifying the limitations of the dispersion conversion required by these types of approaches; since the only alternative in the few cycle limit may be to resort to the much more computationally intensive full Maxwell equation solvers.